A generalized Möbius transform of games on MV-algebras and its application to a Cimmino-type algorithm for the core (Q2906487)

It is well known that any MV-algebra is isomorphic to a separating clan of continuous \([0,1]\)-valued functions over a compact Hausdorff space \(X\). A game on such a clan \(M_X\) is simply a function \(v\) mapping \(M_X\) to the reals, such that \(v\) is bounded and \(v(0) =0\). The author generalises Rota's notion of Möbius transform to the case of games on semisimple MV-algebras. He then characterises the class of games having the generalised Möbius transform as precisely those games arising as the Choquet integral with respect to a capacity, which in turns arises as the difference of two totally monotone capacities on the set of all compact subsets (topologised with the Hausdorff metric topology) of a second countable compact Hausdorff space \(X\). The assumption of second-countability of the space \(X\) is instrumental to the proof of the above characterisation. In the conclusion, the author poses the question whether this assumption can be relaxed.NEWLINENEWLINEThe core of a game \(v\) on \(M_X\) is the set of all bounded measures \(m\) on \(M_X\) such that \(m(1) = v(1)\) and \(m(a) \geq v(a)\) for all \(a \in M_X \setminus \{1\}\). The intuition behind the notion of core is that, in the quoted words of Shapley, ``the core is the set of feasible outcomes that cannot be improved upon by any coalition of players'', where a player is an element of \(X\), and each element of \(M_X\) is a coalition. A bounded measure on \(M_X\) is then a distribution of profit among players. Under some rationality assumptions, such as the one stating that no coalition will accept a smaller profit distribution than the one generated by its own members, an empty core indicates that no coalitions are able to arrive at any agreement about the joint distribution of profits. The generalised Möbius transform is applied to devise an algorithm to test whether the core of a game with finitely many players is empty or not. In the latter case, the algorithm yields a member of the core. The algorithm runs starting from any bounded measure and producing iteratively a sequence of measures whose convergence speed to a solution can be gauged by an auxiliary computable function, which can be tested to check whether the sequence converges to a measure in the core or not.NEWLINENEWLINEFor the entire collection see [Zbl 1241.00017].